The present invention relates to a method and an arrangement for determining at least one digital signal from an electrical signal.
The goal of the information theory established by Claude Shannon in 1948 is to develop efficient codes for encoding transmission and decoding of digital data and to optimally utilize the available information of the encoded data in the decoding insofar as possible.
Yu-Li You et al., “Blind Equalization by Alternating Minimization for Applications to Mobilecommunications”, Globecom 95, IEEE Global Telecommunications Conference, Singapore, Nov. 14-16, 1995, Vol. 1, pp. 88-92, discloses an identification of a transmission channel for the transmission of digital data.
Two types of decoding are distinguished in the decoding of digital data:                in what is referred to as hard decision decoding, a received signal infested with noise by the transmission over a channel is decoded into a sequence of digital data, whereby only the digital value of the respectively received signal is classified; and        in what is referred to as soft decision decoding, an a posteriori probability for the value to be classified is additionally determined for each information character to be decoded. Such a posteriori probabilities are also referred to as soft outputs and form a criterion for the dependability of the decoding.        
Soft decision decoding shall be considered below.
Fundamentals of what are referred to as block codes are known from B. Friedrichs. “Kanalcodierung Grundlagen und Anwendungen in modernen Kommunikationssystemen”, Springer-Verlag, 1996, pp. 69-125, 193-242.
It is known from J. Hagenauer et al., “Iterative Decoding of Binary Block and Convolutional Codes”, IEEE Trans. on Information Theory, Vol. 42, 1996, to implement a soft decision decoding for a binary, linear block code.
The method from J. Hagenauer et al., “Iterative Decoding of Binary Block and Convolutional Codes” for exact calculation of digital signal values from an electrical signal shall be explained below upon employment of what is referred to as log-likelihood algebra.
It is assumed below that the output of a source encoder of a first arrangement is composed of a sequence of digital, preferably binary signal words that are referred to below as code words. The finite number of stochastically independent random variablesUi: Ω→[±1], i=1, . . . , m, mεN  (1) is considered, these being defined on a likelihood space (Ω, S, P). S references a σ-algebra, i.e. the set of events for which a likelihood is defined. P references a likelihood criterion (P:S→[0, 1]). Under the assumption that the inequalities0<P ({ωεΩ;Ui(ω)=01})<1, i=1, . . . , m  (2) are met, what are referred to L-values of the random variables Ui are defined by                                           L            ⁡                          (                              U                i                            )                                :=                      ln            ⁡                          (                                                P                  ⁡                                      (                                          {                                                                        ω                          ∈                          Ω                                                ;                                                                                                            U                              i                                                        ⁡                                                          (                              ω                              )                                                                                =                                                      +                            1                                                                                              }                                        )                                                                    P                  ⁡                                      (                                          {                                                                        ω                          ∈                          Ω                                                ;                                                                                                            U                              i                                                        ⁡                                                          (                              ω                              )                                                                                =                                                      -                            1                                                                                              }                                        )                                                              )                                      ,                  i          =          1                ,        …        ⁢                                   ,                  m          .                                    (        3        )            
Code words u have the following structure:uε{±1}k. 
It is thereby assumed for each code word u that each digital value ui, i=1 . . . k of the code word u assumes a first value (logical “0” or logical “+1”) or a second value (logical “1” or logical “−1”) with the same likelihood. Since one must count on disturbances in the transmission of messages that can falsify the messages, a further encoding step, channel encoding, is implemented.
As described in B. Friedrichs, “Kanalcodlerung Grundlagen und Anwendungen in modemen Kommunikationssystemen”, Springer-Verlag, 1996, pp. 1-30, redundancy is intentionally added to the incoming code words u in the channel encoding in order to be able to correct possible transmission errors and, thus, assure a high transmission dependability. It is assumed below that a channel code word cε{±1}n, n>k, nεN, is allocated to each code word uε{±1}k. The output of the means for channel encoding is thus composed of code words having the form cε{±1}n.
The channel code words are transmitted from a transmission means to a reception means via a physical channel, for example a subscriber line, coaxial cable, mobile radio telephone, directional radio, etc.
Since the physical channel can often not transmit discrete symbols but only time-continuous signals (i.e., specific functions s: →), a modulator is often provided with which a function which is suitable for the transmission via the physical channel, is allocated to the channel code word c. An important characteristic quantity of the transmitted electrical signal is the average energy Eb that is employed for the transmission of an information bit of the channel code word c.
Since a disturbance can occur in the transmission of an electrical signal via a physical channel, an electrical signal {tilde over (s)}: →that is modified compared to the transmitted electrical signal is received.
The disturbance is described with methods of stochastic signal theory. A characteristic quantity of the disturbance is the known single-side noise power density No that is determined by the channel. After a potential demodulation of the received electrical signal {tilde over (s)}, a vector y ε is present instead of the code word c. The absolute amount of each component of the vector y is thereby interpreted as dependability information for the corresponding operational sign of the component in the framework of the soft decision decoding.
The channel decoding then has the job—upon employment of the received, potentially demodulated electrical signal {tilde over (s)} that is ultimately available as vector y—of reconstructing the code word u that was originally present.
It is standard to model the physical channel and the noise properties thereof. A model frequently employed for this purpose is what is referred to as the invariant AWGN channel (additive Gaussian white noise). When a modulator and a demodulator are present the totality of modulator, physical channel and demodulator is referred to below as the channel in this model. Given the AWGN channel, it is assumed that the output of the channel encoder, i.e. the channel code word c is additively superimposed by an   N  ⁡      (                  Q        ⁢                                            N              0                        ⁢            n                                2            ⁢                          E              b                        ⁢            k                              ,              L        n              )  —normally distributed random variable, whereby ln references the n-dimensional unit matrix. The quotient       N    0        E    b  is known and is also referred to as signal-to-noise ratio.
By complete induction for m, it can be shown on the basis of the stochastic independence of the random variables U1, . . . , Um that the following is valid for the L-value of the chained random variables U1⊕ . . . ⊕Um (⊕ references an exclusive-OR operation):U1⊕U2⊕ . . . Um: Ω→{±1},ω→U1(ω)⊕U2(ω)⊕ . . . Um(ω)  (4) 
and                               L          ⁡                      (                                          U                1                            ⊕                              U                2                            ⊕                              …                ⁢                                                                   ⁢                                  U                  m                                                      )                          =                              ln            (                                                                                5                                                                                                              1                      +                                                                        ∏                                                      i                            =                            1                                                    m                                                ⁢                                                                                                   ⁢                                                                                                            exp                              ⁡                                                              (                                                                  L                                  ⁡                                                                      (                                                                          U                                      i                                                                        )                                                                                                  )                                                                                      -                            1                                                                                                              exp                              ⁡                                                              (                                                                  L                                  ⁡                                                                      (                                                                          U                                      i                                                                        )                                                                                                  )                                                                                      +                            1                                                                                                                                                                          1                -                                                      ∏                                          i                      =                      1                                        m                                    ⁢                                                                           ⁢                                                                                    exp                        ⁡                                                  (                                                      L                            ⁡                                                          (                                                              U                                i                                                            )                                                                                )                                                                    -                      1                                                                                      exp                        ⁡                                                  (                                                      L                            ⁡                                                          (                                                              U                                i                                                            )                                                                                )                                                                    +                      1                                                                                            )                    .                                    (        5        )            
The following initial situation derives for the method known from J. Hagenauer et al., “Iterative Decoding of Binary Block and Convolutional Codes natural number k, n and sets Jk+1, . . . , Jn⊂{1, . . . , k}, that describe the properties of the channel encoder are established, as is the non-negative, real number             N      0              E      b        .The number of digital values of the channel code word u is referenced k. The number of digital values of the channel code word cε{±1}n, is referenced n, with n>k. The n-k digital values that are attached to the code words u in the formation of the channel code word c, which are also referred to as check bits, are characterized by Jk+1, . . . , Jn⊂{1, . . . , k}.
Further, a likelihood space (Ω, S, P) and a small-dimensional random variable CCΩ→{±1}n  (6) having the following properties is established:                componentsC1, . . . ,Ck:Ω→{±1}  (7) of the n-dimensional random variable C are stochastically independent andP(ωεΩ;Ci(ω)=−1)=P(ωεΩ;Ci(ω)=+1)=½  (8) applies to all i=1, . . . , k.        the following applies to each iε{k+1, . . . , n} and to all ωεΩ:                                           C            i                    ⁡                      (            ω            )                          =                              ⊕                          j              ⁢                                                           ⁢              ɛ              ⁢                                                           ⁢                              J                i                                              ⁢                                                    C                j                            ⁡                              (                ω                )                                      .                                              (        9        )                    
The digital values that are formed by the channel encoding, i.e. the channel code words c, are interpreted as realization of the random variables C.
The output ū of the channel decoder to be reconstructed, which is referred to below as a set of digital signal values, are the corresponding realization of the random variablesU: Ω→{±1}k,ω→(C1(ω), . . . , Ck(ω))T  (10). The outputyεn  (11) of the unit for demodulation or, the vector that describes the electrical signal and for which the decoding ensues is interpreted as realization of the random variablesY:→n, ω→C(ω)+Z(ω)  (12) whereby Z: Ω→n is   N  ⁡      (                  Q        ⁢                                            N              0                        ⁢            n                                2            ⁢                          E              b                        ⁢            k                              ,              L        n              )  —normally distributed random variable that is stochastically independent of the n-dimensional random variable C. The code word ũ is reconstructed based on the vector y describing the received electrical signal.
In order to reconstruct the individual digital signal values, the distribution of the random variables C is investigated under the condition that the vector y describing the electrical signal was received.
The likelihoods induced by this distribution are referred to as a posteriori likelihoods.
The following quantities are considered for each ε>0:                                                                                           L                  ɛ                                ⁡                                  (                                                            U                      i                                        |                                          y                      _                                                        )                                            :=                            ⁢                                                ln                  ⁢                                      (                                                                  P                        ⁡                                                  (                                                                                    {                                                                                                ω                                  ∈                                  Ω                                                                ;                                                                                                                                            U                                      i                                                                        ⁡                                                                          (                                      ω                                      )                                                                                                        =                                                                      +                                    1                                                                                                                              }                                                        |                                                          {                                                                                                ω                                  ∈                                  Ω                                                                ;                                                                                                                                            Y                                      _                                                                        ⁡                                                                          (                                      ω                                      )                                                                                                        ∈                                                                      M                                                                                                                  y                                        _                                                                            ,                                      ɛ                                                                                                                                                                  }                                                                                )                                                                                            P                        ⁡                                                  (                                                                                    {                                                                                                ω                                  ∈                                  Ω                                                                ;                                                                                                                                            U                                      i                                                                        ⁡                                                                          (                                      ω                                      )                                                                                                        =                                                                      +                                    1                                                                                                                              }                                                        |                                                          {                                                                                                ω                                  ∈                                  Ω                                                                ;                                                                                                                                            Y                                      _                                                                        ⁡                                                                          (                                      ω                                      )                                                                                                        ∈                                                                      M                                                                                                                  y                                        _                                                                            ,                                      ɛ                                                                                                                                                                  }                                                                                )                                                                                      )                                                  =                                                                                        =                            ⁢                                                ln                  ⁡                                      (                                                                                            ∑                                                                                                                    v                                _                                                            ∈                              C                                                                                                                      v                                i                                                            =                                                              +                                1                                                                                                                                    ⁢                                                  P                          ⁡                                                      (                                                                                          {                                                                                                      ω                                    ∈                                    Ω                                                                    ;                                                                                                                                                    C                                        _                                                                            ⁡                                                                              (                                        ω                                        )                                                                                                              =                                                                          v                                      _                                                                                                                                      }                                                            |                                                              {                                                                                                      ω                                    ∈                                    Ω                                                                    ;                                                                                                                                                    Y                                        _                                                                            ⁡                                                                              (                                        ω                                        )                                                                                                              ∈                                                                          M                                                                                                                        y                                          _                                                                                ,                                        ɛ                                                                                                                                                                            }                                                                                      )                                                                                                                                                ∑                                                                                                                    v                                _                                                            ∈                              C                                                                                                                      v                                i                                                            =                                                              -                                1                                                                                                                                    ⁢                                                  P                          ⁡                                                      (                                                                                          {                                                                                                      ω                                    ∈                                    Ω                                                                    ;                                                                                                                                                    C                                        _                                                                            ⁡                                                                              (                                        ω                                        )                                                                                                              =                                                                          v                                      _                                                                                                                                      }                                                            |                                                              {                                                                                                      ω                                    ∈                                    Ω                                                                    ;                                                                                                                                                    Y                                        _                                                                            ⁡                                                                              (                                        ω                                        )                                                                                                              ∈                                                                          M                                                                                                                        y                                          _                                                                                ,                                        ɛ                                                                                                                                                                            }                                                                                      )                                                                                                                )                                                  .                                                                        (        13        )            for i=1, . . . , k, wherebyMy,ε:=[y1,y1+ε]x . . . x[yn,yn+ε]  (14) and C references the set of all channel code words c.
The following derives by employing theorem of Bayes:                                                                                           L                  ɛ                                ⁡                                  (                                                            U                      i                                        |                                          y                      _                                                        )                                            :=                            ⁢                              ln                ⁡                                  (                                                                                    ∑                                                                                                            v                              _                                                        ∈                            C                                                                                                              v                              i                                                        =                                                          +                              1                                                                                                                          ⁢                                              P                        ⁡                                                  (                                                                                    {                                                                                                ω                                  ∈                                  Ω                                                                ;                                                                                                                                            Y                                      _                                                                        ⁡                                                                          (                                      ω                                      )                                                                                                        ∈                                                                      M                                                                                                                  y                                        _                                                                            ,                                      ɛ                                                                                                                                                                  }                                                        |                                                          {                                                                                                ω                                  ∈                                  Ω                                                                ;                                                                                                                                            C                                      _                                                                        ⁡                                                                          (                                      ω                                      )                                                                                                        =                                                                      v                                    _                                                                                                                              }                                                                                )                                                                                                                                    ∑                                                                                                            v                              _                                                        ∈                            C                                                                                                              v                              i                                                        =                                                          -                              1                                                                                                                          ⁢                                              P                        ⁡                                                  (                                                                                    {                                                                                                ω                                  ∈                                  Ω                                                                ;                                                                                                                                            Y                                      _                                                                        ⁡                                                                          (                                      ω                                      )                                                                                                        ∈                                                                      M                                                                                                                  y                                        _                                                                            ,                                      ɛ                                                                                                                                                                  }                                                        |                                                          {                                                                                                ω                                  ∈                                  Ω                                                                ;                                                                                                                                            C                                      _                                                                        ⁡                                                                          (                                      ω                                      )                                                                                                        =                                                                      v                                    _                                                                                                                              }                                                                                )                                                                                                      )                                                                                                        =                            ⁢                                                ln                  ⁡                                      (                                                                                            ∑                                                                                                                    v                                _                                                            ∈                              C                                                                                                                      v                                i                                                            =                                                              +                                1                                                                                                                                    ⁢                                                                              ∫                                                          M                                                                                                y                                  _                                                                ,                                ɛ                                                                                                              ⁢                                                                                    exp                              (                                                              -                                                                                                                                                                                    (                                                                                                                              x                                            _                                                                                    -                                                                                      v                                            _                                                                                                                          )                                                                            T                                                                        ⁢                                                                          (                                                                                                                        x                                          _                                                                                -                                                                                  v                                          _                                                                                                                    )                                                                                                                                                                                                                          N                                        0                                                                            ⁢                                      n                                                                                                                                                      E                                        b                                                                            ⁢                                      k                                                                                                                                                                  )                                                        ⁢                                                          ⅆ                              x                                                                                                                                                                            ∑                                                                                                                    v                                _                                                            ∈                              C                                                                                                                      v                                i                                                            =                                                              -                                1                                                                                                                                    ⁢                                                                              ∫                                                          M                                                                                                y                                  _                                                                ,                                ɛ                                                                                                              ⁢                                                                                    exp                              (                                                              -                                                                                                                                                                                    (                                                                                                                              x                                            _                                                                                    -                                                                                      v                                            _                                                                                                                          )                                                                            T                                                                        ⁢                                                                          (                                                                                                                        x                                          _                                                                                -                                                                                  v                                          _                                                                                                                    )                                                                                                                                                                                                                          N                                        0                                                                            ⁢                                      n                                                                                                                                                      E                                        b                                                                            ⁢                                      k                                                                                                                                                                  )                                                        ⁢                                                          ⅆ                              x                                                                                                                                            )                                                  .                                                                        (        15        )            
When the boundary transition of (14) for ε→0 is considered by multiple employment of the rule of De L'Hospital, then the soft outputs L(Ui|y) are obtained for each character according to the following rule:                               L          ⁡                      (                                          U                i                            |                              y                _                                      )                          =                              ln            ⁡                          (                                                                    ∑                                                                                            v                          _                                                ∈                        C                                                                                              v                          i                                                =                                                  +                          1                                                                                                      ⁢                                      exp                    (                                          -                                                                                                                                  (                                                                                                y                                  _                                                                -                                                                  v                                  _                                                                                            )                                                        T                                                    ⁢                                                      (                                                                                          y                                _                                                            -                                                              v                                _                                                                                      )                                                                                                                                                              N                              0                                                        ⁢                            n                                                                                                              E                              b                                                        ⁢                            k                                                                                                                )                                                                                        ∑                                                                                            v                          _                                                ∈                        C                                                                                              v                          i                                                =                                                  -                          1                                                                                                      ⁢                                      exp                    (                                          -                                                                                                                                  (                                                                                                y                                  _                                                                -                                                                  v                                  _                                                                                            )                                                        T                                                    ⁢                                                      (                                                                                          y                                _                                                            -                                                              v                                _                                                                                      )                                                                                                                                                              N                              0                                                        ⁢                            n                                                                                                              E                              b                                                        ⁢                            k                                                                                                                )                                                              )                                .                                    (        16        )            
The soft outputs that, on the one hand, usually contain an operational sign information and a dependability information (absolute amount of the soft output), are referred to below as a dependability degree.
In a completely analogous way, the following is obtained for i=k+1, . . . , n:                               L          ⁡                      (                                                            ⊕                                      j                    ∈                                          J                      i                                                                      ⁢                                  U                  j                                            |                              y                _                                      )                          =                              ln            ⁡                          (                                                                    ∑                                                                                            v                          _                                                ∈                        C                                                                                              v                          i                                                =                                                  +                          1                                                                                                      ⁢                                      exp                    (                                          -                                                                                                                                  (                                                                                                y                                  _                                                                -                                                                  v                                  _                                                                                            )                                                        T                                                    ⁢                                                      (                                                                                          y                                _                                                            -                                                              v                                _                                                                                      )                                                                                                                                                              N                              0                                                        ⁢                            n                                                                                                              E                              b                                                        ⁢                            k                                                                                                                )                                                                                        ∑                                                                                            v                          _                                                ∈                        C                                                                                              v                          i                                                =                                                  -                          1                                                                                                      ⁢                                      exp                    (                                          -                                                                                                                                  (                                                                                                y                                  _                                                                -                                                                  v                                  _                                                                                            )                                                        T                                                    ⁢                                                      (                                                                                          y                                _                                                            -                                                              v                                _                                                                                      )                                                                                                                                                              N                              0                                                        ⁢                            n                                                                                                              E                              b                                                        ⁢                            k                                                                                                                )                                                              )                                .                                    (        17        )            
The decoding in the known method ensues such that, when the dependability degree exhibits a value greater than 0, the ith component ui of the code word ũ to be reconstructed is reconstructed with the second value (logical “1” or logical “−1”). For a value of the dependability degree less than 0, the first value (logical “0” or logical “+1”) is allocated to the digital signal value. One can arbitrarily decide in favor of the first or the second value for a value of the dependability degree equal to 0. The absolute amount of the dependability degree is a criterion for the dependability of the above decision rules. The reconstruction is all the more dependable the higher the absolute amount.
What is disadvantageous about this known method is the outlay for computer-assisted determination of the dependability degree. The determination of the dependability degree generally requires an outlay for additions that is proportional to min (2k, 2n−k). The direct calculation of the dependability degrees and the determination of the digital values dependent on the dependability degrees is thus often not numerically realizable. Approximately 1020 additions would be required for what is referred to as the BCH (255, 191)—Code (see B. Friedrichs, Kanalcodierung Grundlagen und Anwendungen in moderen Kommunikationssystemen”, Springer-Verlag, 1996, pp. 69-125, 193-242, for the calculation of the 191 dependability degrees and digital signal values.